Problem: $g(x)=-5x^4+4x^3-20x-20$. On which intervals is the graph of $g$ concave up? Choose 1 answer: Choose 1 answer: (Choice A) A $x<-\dfrac52$ and $x>0$ (Choice B) B $0<x<\dfrac{2}{5}$ only (Choice C) C $x>5$ only (Choice D) D $x<-2$ and $x>-\dfrac{2}{3}$
Solution: We can analyze the intervals where $g$ is concave up/down by looking for the intervals where its second derivative $g''$ is positive/negative. This analysis is very similar to finding increasing/decreasing intervals, only instead of analyzing $g'$, we are analyzing $g''$. The second derivative of $g$ is $g''(x)=-12(x)(5x-2)$. $g''(x)=0$ for $x=0,\dfrac{2}{5}$. Since $g''$ is a polynomial, it's defined for all real numbers. Therefore, our points of interest are $x=0$ and $x=\dfrac{2}{5}$. Our points of interest divide the number line into three intervals: $\llap{-}2$ $\llap{-}1$ $0$ $1$ $2$ $x<0$ $0<x<\frac25$ $x>\frac25$ Let's evaluate $g''$ at each interval to see if it's positive or negative on that interval. Interval $x$ -value $g''(x)$ Verdict $x<0$ $x=-1$ $g''(-1)=-84<0$ $g$ is concave down $\cap$ $0<x<\dfrac25$ $x=\dfrac15$ $g''\left(\dfrac15\right)=\dfrac{12}5>0$ $g$ is concave up $\cup$ $x>\dfrac25$ $x=1$ $g''(1)=-36<0$ $g$ is concave down $\cap$ In conclusion, the graph of $g$ is concave up over the interval $0<x<\dfrac{2}{5}$ only.